3. Continuous Random Variables

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Uniform distribution The continuous random variable has the Uniform distribution between and , with if { , for short. Statistics and probability: 3-2 Roughly speaking, , if X can only take values between and , and any value of within these values is as likely as any other value. Mean and variance: for , Proof: Let y be the distance from the mid-point, , and the width be . Then since means add 〈 〉 〈 〉 Unsurprisingly the mean is the midpoint. ∫ Occurrence of the Uniform distribution ∫ 1) Waiting times from random arrival time until a regular event (see below) 2) Engineering tolerances: e.g. if a diameter is quoted "�0.1mm", it sometimes assumed (probably incorrectly) that the error has a U(-0.1, 0.1) distribution. 3) Simulation: programming languages often have a standard routine for simulating the U(0, 1) distribution. This can be used to simulate other probability distributions. f(x) and ∫ * + � � 1 2 ( ) x
Example: Disk wait times Statistics and probability: 3-3 In a hard disk drive, the disk rotates at 7200rpm. The wait time is defined as the time between the read/write head moving into position and the beginning of the required information appearing under the head. (a) Find the distribution of the wait time. (b) Find the mean and standard deviation of the wait time. (c) Booting a computer requires that 2000 pieces of information are read from random positions. What is the total expected contribution of the wait time to the boot time, and rms deviation? Solution Rotation time = 8.33ms. Wait time can be anything between 0 and 8.33ms and each time in this range is as likely as any other time. Therefore, distribution of the wait time is U(0, 8.33ms) (i. . � 1 = 0 and � 2 = 8.33ms). For 2000 reads the mean time is 2000 4.2 ms = 8.3s. The variance is 2000 5.7ms 2 = 0.012s 2 , so � =0.11s. Exponential distribution The continuous random variable has the Exponential distribution, parameter if: { ; Relation to Poisson distribution: If a Poisson process has constant rate , the mean after a time is . The probability of no-occurrences in this time is
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Page 5 and 6: ∫ ∫ (b) Mean = = 9491 hours. St
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Page 9 and 10: i.e. the required range is 14.96mm
Page 11 and 12: Normal approximation to the Binomia
Page 13: Example: Stock Control Statistics a

3. Continuous Random Variables

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